Why Option-Implied Risk-Neutral Probabilities Differ from Real-World Probabilities
Summary
The document explains why option prices can be used to infer a risk-neutral probability distribution, but do not directly reveal real-world probabilities. The risk-neutral measure combines physical probabilities with the stochastic discount factor, which weights outcomes according to their pricing and marginal-utility implications. Because both components are unknown, option prices alone do not identify the physical distribution.
One possible route is to assume a particular discount-factor model, such as one based on power utility, and derive probabilities under that assumption. The note cautions that misspecifying the model can produce misleading estimates. It also describes recovery theory as an active research area, while citing doubts about whether a proposed recovery result matches realized returns and variances. Conceptually, risk-neutral distributions may assign more weight to adverse outcomes that investors especially fear, and less to favorable outcomes, though the actual difference depends on the pricing model. The document offers no empirical distribution or estimation procedure and leaves recovery unresolved.
Key ideas
- Option prices imply a risk-neutral distribution rather than directly revealing real-world probabilities.
- The risk-neutral measure reflects both physical likelihoods and the stochastic discount factor.
- Recovering physical probabilities requires assumptions about the discount factor or additional theory.
- A misspecified pricing model can lead to unreliable real-world probability estimates.
- Risk-neutral weighting can emphasize adverse states that carry high marginal utility.
Tags
Full text
# Real world probabilities from option implied risk neutral density?
# Real world probabilities from option implied risk neutral density?
The work of Breeden and Litzenberger-formula (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2642349) gives us a risk neutral probability distribution of a stock price, depending on the option prices.
Question 1: Is there a theoretical or practical way of obtaining real world probabilities given risk neutral ones?
Question 2: What does a real world probability distribution look like compared to a risk neutral, in empirical cases?
## Answer by Kevin (score 5, accepted)
https://quant.stackexchange.com/a/61496
Great question! Unfortunately, it's not easy. We can use option prices to get the $\mathbb{Q}$-distribution. However, the probability measure $\mathbb{Q}$ merges the stochastic discount factor (SDF) $M$ and the real world probabilities, $\mathbb{P}$, and it's not clear how to untangle the two (see this answer). Essentially, you have one equation, but two unknowns.
You can recover $\mathbb{P}$ if you make assumptions about the SDF $M$, see this answer for an example (power utility). However, the asset pricing literature has many, many different models for the SDF and a misspecified $M$ gives you misspecified real world probabilities...
There is hope though. ``Recovery theory'' is part of current research. Famously, Steve Ross proposed a possibility in his 2015 JF publication. However, Jackwerth and Menner (2020, JFE) cast doubt whether the recovery theorem is compatible with future realised returns and variances. So, this recovery is still being researched.
How do $\mathbb{Q}$ and $\mathbb{P}$ differ? Well, the answer is the SDF. You can take the simplest models (log-normal) to get a taste, see this answer. Essentially, the likelihood of bad events get inflated under $\mathbb{Q}$ because these are the states with high marginal utility (= that investors fear) whereas $\mathbb{Q}$ puts less weight on good events.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.